If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $\tau$ is a unitary representation of $H$, then the authors give a sufficient condition on $\xi \in i{\mathfrak{g}}^{\ast }$ to be in the wave front set of ${Ind}_H^G\tau$. In the special case where $\tau$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $\pi$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $WF\left(\pi \right)$ in terms of the direct integral decomposition of $\pi$ into irreducibles. Special cases of this result were previously obtained by Kashiwara–Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.

We prove two main results. (1) Suppose that $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb{C}}^m$ asymptotic at infinity to the union ${\Pi }_1\cup {\Pi }_2$ of two transverse special Lagrangian planes ${\Pi }_1,{\Pi }_2$ in ${\mathbb{C}}^m$ for $m\ge 3$. Then $L$ is one of the explicit Lawlor neck family of examples found by Lawlor. (2) Suppose that $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb{C}}^m$ asymptotic at infinity to the union ${\Pi }_1\cup {\Pi }_2$ of two transverse Lagrangian planes ${\Pi }_1,{\Pi }_2$ in ${\mathbb{C}}^m$ for $m\ge 3$. Then $L$ is one of the explicit family of examples in recent work by Joyce, Lee, and Tsui. If instead $L$ is immersed rather than embedded, the only extra possibility in (1), (2) is $L={\Pi }_1\cup {\Pi }_2$. Our methods, which are new and can probably be used to prove other similar uniqueness theorems, involve $J$-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories from symplectic topology.

Motivated by the recent work of Hassainia and Hmidi, we close the question of the existence of convex global rotating solutions for the generalized surface quasi-geostrophic equation for $\alpha \in [1,2)$. We also show $C^{\infty }$-regularity of their boundary for all $\alpha \in (0,2)$.